Copyright by Athena Ranice Stacy 2011 - The University of Texas at ...

with mass and accretion rate is quite similar to the results of these more
detailed previous studies, and here serves to highlight the relevant physics.
During the next phase, the protostellar radius begins to shrink due to
Kelvin-Helmholtz contraction. The major source of opacity in the interior of
the protostar is electron scattering, which is independent of density and tem-
perature. This along with hydrostatic equilibrium yields the mass-luminosity
relation L∗ ∝ M 3 ∗ . Using figure 4 of Omukai and Palla (2003), we normalize
this relation as follows:
L∗II 10 4 L⊙
M∗
10 M⊙
3
, (2.13)
where L∗II is the protostellar luminosity during the Kelvin-Helmholtz contrac-
tion phase. If we further assume that tKH tacc, we arrive at
GM 2 ∗
R∗L∗
M∗
˙M
. (2.14)
Using L∗ L∗II, we now find a relation for R∗II, the protostellar radius during
the Kelvin-Helmholtz contraction phase:
R∗II 140R⊙
M˙
M∗
−2 Mfid
˙ 10M⊙
. (2.15)
In determining the evolution of Lacc, we set R∗ = R∗I at early times
when the protostar is still experiencing adiabatic accretion. When M∗ ∼ 10 M⊙,
after approximately 800 yr of accretion, the value of R∗II falls below that of
R∗I. At this point, we estimate that Kelvin-Helmholtz contraction has com-
menced and switch to setting R∗ = R∗II. The resulting transition in the
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